Derivada de (x*tgx)/(1-cos^2x+tgx^2)

Ha introducido [src]

       x*tan(x)      
---------------------
       2         2   
1 - cos (x) + tan (x)

\frac{x \tan{\left(x \right)}}{\left(1 - \cos^{2}{\left(x \right)}\right) + \tan^{2}{\left(x \right)}}

(x*tan(x))/(1 - cos(x)^2 + tan(x)^2)

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x \tan{\left(x \right)}$ y $g{\left(x \right)} = - \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1$ miembro por miembro:
  1. La derivada de una constante $1$ es igual a cero.
  2. Sustituimos $u = \tan{\left(x \right)}$ .
  3. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
    1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
    Como resultado de la secuencia de reglas:
    
    $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  5. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
    1. Sustituimos $u = \cos{\left(x \right)}$ .
    2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Como resultado de la secuencia de reglas:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    Entonces, como resultado: $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Como resultado de: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- x \left(\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \tan{\left(x \right)} + \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \left(- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{\left(- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)^{2}}$
Simplificamos:

$\frac{- 2 x \left(- \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right) + \left(x + \frac{\sin{\left(2 x \right)}}{2}\right) \left(\sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right)}{\left(\sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right)^{2} \cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{- 2 x \left(- \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right) + \left(x + \frac{\sin{\left(2 x \right)}}{2}\right) \left(\sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right)}{\left(\sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}\right)^{2} \cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

  /       2   \              /  /         2   \                         \       
x*\1 + tan (x)/ + tan(x)   x*\- \2 + 2*tan (x)/*tan(x) - 2*cos(x)*sin(x)/*tan(x)
------------------------ + -----------------------------------------------------
        2         2                                              2              
 1 - cos (x) + tan (x)                    /       2         2   \               
                                          \1 - cos (x) + tan (x)/

\frac{x \left(- \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \tan{\left(x \right)}}{\left(\left(1 - \cos^{2}{\left(x \right)}\right) + \tan^{2}{\left(x \right)}\right)^{2}} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{\left(1 - \cos^{2}{\left(x \right)}\right) + \tan^{2}{\left(x \right)}}

Simplificar

Segunda derivada [src]

  /                                                                                                               /                                                                             2                          \       \
  |                                                                                                               |             2                         //       2   \                       \                           |       |
  |                                                                                                               |/       2   \       2         2      4*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/         2    /       2   \|       |
  |                                                                                                             x*|\1 + tan (x)/  + cos (x) - sin (x) - ----------------------------------------- + 2*tan (x)*\1 + tan (x)/|*tan(x)|
  |                                         /  /       2   \         \ //       2   \                       \     |                                                      2         2                                       |       |
  |       2        /       2   \          2*\x*\1 + tan (x)/ + tan(x)/*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/     \                                               1 + tan (x) - cos (x)                                    /       |
2*|1 + tan (x) + x*\1 + tan (x)/*tan(x) - ------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------|
  |                                                                     2         2                                                                                   2         2                                                  |
  \                                                              1 + tan (x) - cos (x)                                                                         1 + tan (x) - cos (x)                                               /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                              2         2                                                                                                           
                                                                                                       1 + tan (x) - cos (x)

\frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{x \left(- \frac{4 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2}}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} - \frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} + \tan^{2}{\left(x \right)} + 1\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}

Simplificar

Tercera derivada [src]

  /                                                                                                                                                                                                                                                                               /                                                                                                          3                                            /             2                                              \\       \
  |                                                                                                                                                              /                                                                             2                          \       |                                                       2            //       2   \                       \      //       2   \                       \ |/       2   \       2         2           2    /       2   \||       |
  |                                                                                                                                                              |             2                         //       2   \                       \                           |       |   3    /       2   \                     /       2   \           6*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/    3*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/*\\1 + tan (x)/  + cos (x) - sin (x) + 2*tan (x)*\1 + tan (x)//|       |
  |                                                                                                                                   /  /       2   \         \ |/       2   \       2         2      4*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/         2    /       2   \|   4*x*|tan (x)*\1 + tan (x)/ - cos(x)*sin(x) + 2*\1 + tan (x)/ *tan(x) + ----------------------------------------- - -------------------------------------------------------------------------------------------------------|*tan(x)|
  |                                                                                                                                 3*\x*\1 + tan (x)/ + tan(x)/*|\1 + tan (x)/  + cos (x) - sin (x) - ----------------------------------------- + 2*tan (x)*\1 + tan (x)/|       |                                                                                                  2                                                           2         2                                            |       |
  |                                                 //       2   \                       \ /       2        /       2   \       \                                |                                                      2         2                                       |       |                                                                           /       2         2   \                                                     1 + tan (x) - cos (x)                                         |       |
  |/       2   \ /             /         2   \\   6*\\1 + tan (x)/*tan(x) + cos(x)*sin(x)/*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/                                \                                               1 + tan (x) - cos (x)                                    /       \                                                                           \1 + tan (x) - cos (x)/                                                                                                                   /       |
2*|\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)// - ------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
  |                                                                                   2         2                                                                                                   2         2                                                                                                                                                                            2         2                                                                                                          |
  \                                                                            1 + tan (x) - cos (x)                                                                                         1 + tan (x) - cos (x)                                                                                                                                                                  1 + tan (x) - cos (x)                                                                                                       /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                     2         2                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                              1 + tan (x) - cos (x)

\frac{2 \left(- \frac{4 x \left(\frac{6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right)^{3}}{\left(- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)^{2}} - \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)}\right) \tan{\left(x \right)}}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} - \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(- \frac{4 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right)^{2}}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1} + \left(x \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)}\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}\right)}{- \cos^{2}{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}

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Derivada de (x*tgx)/(1-cos^2x+tgx^2)

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